Ordinary differential cohomology is the differential cohomology-refinement of ordinary cohomology, for instance realized as singular cohomology.
Every generalized (Eilenberg-Steenrod) cohomology-theory has a refinement to differential cohomology. By ordinary differential cohomology one refers, for emphasis, to the differential refinement of ordinary integral cohomology , hence of the cohomology theory represented by the Eilenberg-MacLane spectrum $K(-,\mathbb{Z})$. To the extent that integral cohomology is often just called cohomology when the context is clear, ordinary differential cohomology is often called just differential cohomology .
Ordinary differential cohomology classifies circle n-bundles with connection. In low degree these are ordinary circle bundles with connection. In the next degree they are circle 2-group principal 2-bundles / bundle gerbes with 2-connection.
Here we write $H_{diff}^\bullet(X)$ for the ordinary differential cohomology groups of a smooth manifold $X$.
There are two natural morphisms:
The underlying characteristic class
produces the class in integral cohomology that underlies a differential cocycle;
for $H^2_{diff}(X)$ this is called the first Chern class of a line bundle;
for $H^3_{diff}(X)$ this is called the Dixmier-Douady class of the corresponding bundle gerbe.
The curvature
produces a closed differential form of degree $n$. This happens to land in closed differential forms with integral periods (see below).
The following is either a definition, if regarded as an axiomatic characterization of ordinary differential cohomology, or it is a proposition, if regarded as a property of one of the models.
Let $X$ be a smooth manifold and $n \in \mathbb{N}$ with $n \geq 1$ Write
$H_{diff}^n(X)$ for the ordinary differential cohomology of $X$ in degree $n$;
$\Omega^{n-1}(X)$ for the collection of differential forms of degree $n-1$;
$\Omega^{n-1}_{int}(X)$ for the collection of differential forms $\omega$ of degree $n-1$ that are closed and whose periods are integral: for every $\gamma : S^{n-1} \to X$ we have the integral $\int_{S^{n-1}} \gamma^* \omega \in \mathbb{Z} \hookrightarrow \mathbb{R}$. Similarly for $\Omega^n_{int}(X)$.
$H^n(X, \mathbb{Z})$ and $H^n(X, U(1))$ for the ordinary cohomology (for instance modeled as singular cohomology) of $X$ with coefficients in the integers or the circle group (regarded as a discrete group), respectively.
All of these sets are abelian groups: the forms under addition of forms, and the differential cohomology classes are defined or proven (depending on the approach, see above) to have abelian group structure such that the maps to curvatures and characteristic classes, from above are homomorphisms of abelian groups.
The differential cohomology $H_{diff}^n(X)$ of $X$ fits into short exact sequences of abelian groups
curvature exact sequence
characteristic class exact sequence
The first sequence (1) says in words: two circle $(n-1)$-bundles $n$ whose curvature coincides differ by a flat circle (n-1)-bundle.
The second sequence (2) says in words: two connections on the same circle $(n-1)$-bundle differ by a globally defined connection $(n-1)$-form, well defined up to addition of a form with integral periods.
More is true: both these sequences interlock to form the hexagonal differential cohomology diagram of ordinary differential cohomology. For more see at differential cohomology diagram – Examples – Deligne coefficients.
There are various equivalent cocycle-models for ordinary differential cohomology. They include
The last of these are often known as $U(1)$-gerbes or bundle gerbes with connection.
Cocycles $\nabla \in H_{diff}^2(X)$ in degree 2 ordinary differential cohomology are represented by ordinary circle group-principal bundles with connection on $X$. The class $c(\nabla) \in H^2(X,\mathbb{Z})$ is the Chern class of the underlying circle bundle and the form $F_\nabla \in \Omega^2_{cl}(X)$ is the curvature 2-form of the connection $\nabla$.
see
For $X$ a smooth manifold, $G$ a Lie group with Lie algebra $\mathfrak{g}$ and invariant polynomial $\langle -\rangle$ of degree $2n$, the Chern-Weil homomorphism may be refined to a morphism
from the first nonabelian cohomology of $X$ classifying $G$-principal bundles to degree $2n$ ordinary differential cohomology.
The projection
is the integral characteristic class corresponding to the invariant polynomial and the projection
is a differential form which represents the image of this class under $H^{2n}(X,\mathbb{Z}) \to H^{2n}(X,\mathbb{R})$ in de Rham cohomology (under the de Rham theorem).
In physics
the electromagnetic field is a cocycle in degree 2 ordinary differential cohomology
the Kalb-Ramond field is a cocycle in degree 3;
the supergravity C-field is a cocycle in degree 4.
In abelian higher dimensional Chern-Simons theory in dimension $(4k+3)$ a field configuration is a cocycle in ordinary differential geometry of degree $(2k+2)$, for $k \in \mathbb{N}$.
Ordinary differential cohomology (and indeed a cocycle model thereof) is defined generally internal to any cohesive (∞,1)-topos $\mathbf{H}$. This is discussed at
For the case $\mathbf{H} =$ Smooth∞Grpd this intrinsic definition reproduces the Deligne complex model. This is discussed at
For original constructions see the references at
For discussion in the broader context of generalized differential cohomology see
Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology,and M-Theory (arXiv)
A pedestrian introduction of ordinary differential cocycles is in section 2.3 there.
The systematic construction and definition via a homotopy pullback is in section 3.2.
The relation to Chern-Weil theory is in section 3.3.
A characterization by the two characteristic exact sequences is discussed in
Discussion of equivariant ordinary differential cohomology
Ernesto Lupercio, Bernardo Uribe, Deligne Cohomology for Orbifolds, discrete torsion and B-fields, in: Geometric and topological methods for quantum field theory Proceedings, Summer School, Villa de Leyva, Colombia, July 9-27, 2001 (arXiv:hep-th/0201184, spire:582101)
Kiyonori Gomi, Equivariant smooth Deligne cohomology, Osaka J. Math. Volume 42, Number 2 (2005), 309-337 (euclid:ojm/1153494380)
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Byungdo Park, Corbett Redden, A classification of equivariant gerbe connections, Communications in Contemporary MathematicsVol. 21, No. 02, 1850001 (2019) (doi:10.1142/S0219199718500013)
Cheng-Yong Du Xiaojuan Zhao, Spark and Deligne-Beilinson cohomology on orbifolds, Journal of Geometry and Physics Volume 104, June 2016, Pages 277-290 (doi:10.1016/j.geomphys.2016.02.011)
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